For lines, squares, and cubes we have seen there is a relationship between a scaling factor r and the number N(r) of copies of a similar shape, scaled by a factor of r, needed to cover the original shape.
Representing by d the Euclidean dimension of these shapes, the relationship is
N(r) = (1/r)d.
Now we assert that if this scaling relationship holds for a shape, then the value of d is the dimension of the shape.
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